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Journal of Physics C Solid State Physics

A real-space renormalisation group for the triangular lattice

N -C Chao

Published Jun 20, 1981• 4 citations
Publisher

Abstract

The author extends the real-space renormalisation group approach of Reynolds, Klein and Stanley (1977) to the triangular lattice using a three-bond cluster as an elementary cluster. His results for the bond-percolation and Ising fixed points show considerable improvement over those of the decimation results of Yeomans and Stinchcombe (1979) and agree with the exact values to within 11% error.

DOI
10.1088/0022-3719/14/17/002

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Journal of Physics A Mathematical and General•Aug 21, 1989•0 citations
The author extends the real-space renormalisation group approach of Reynolds, Klein and Stanley (1977) to the triangular lattice using a three-bond cluster as an elementary cluster. His results for the bond-percolation and Ising fixed points show considerable improvement over those of the decimation results of Yeomans and Stinchcombe (1979) and agree with the exact values to within 11% error.
The author proposes a new type of cluster in addition to those of Reynolds, Klein and Stanley for the square lattice and another in addition to those of Chao for the triangular lattice for a real-space renormalisation group approach for studying critical phenomena. Thus each lattice is associated with two types of clusters-the longitudinal and the diagonal. The results for the bond percolation and Ising fixed points obtained from the diagonal square and honeycomb clusters agree with the exact values to within 6-11% error; whereas those obtained from the longitudinal triangular and honeycomb clusters agree with the exact values to within 2-4.5% error. The critical exponents corresponding to the percolation and the Ising problems also show reasonably good results: in particular those calculated by using the longitudinal triangular cluster agree with the exact values to within 6.7% and 3.2% error respectively.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
A renormalisation group approach developed for the percolation problem by Young and Stinchcombe (1975) is extended to yield magnetic indices. A real-space cluster calculation is carried out for the Ashkin-Teller-Potts model (Ashkin & Teller 1964, Potts 1952) with general S, allowing contact with results for the Ising model (S to 2) as well as with the previously published thermal indices of the percolation problem (S to 1). For the percolation problem, the magnetic index obtained is 2.06, about 10% larger than current series estimates, and results in gamma =2.87.
Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.